A new derivation of the Henon's isochrone potentials

TL;DR

This paper presents a simplified derivation of Hénon's isochrone potentials using Abel inversion, demonstrating that isochronous radial periods are Keplerian and providing insights into inverse central-force problems and Bertrand's theorem.

Contribution

The authors introduce a straightforward Abel inversion-based method to derive all isochrone central potentials and establish the Keplerian nature of isochronous radial periods.

Findings

Revealed all isochrone central potentials under mild assumptions.

Showed isochronous radial periods are necessarily Keplerian.

Provided a proof of Bertrand's theorem using the new approach.

Abstract

We revisit in this note the H\'enon's isochrone problem. By using the standard Abel inversion technique for one-dimensional motion, we recover in a simple way the H\'enon's parabolae and get all isochrone central potentials under mild smoothness assumptions on the potential function. Our approach also allows us to conclude that isochronous radial periods with explicit energy dependence are necessarily Keplerian, i.e., $T^{2}\propto|E|^{-3}$, and that their corresponding orbits can be easily integrated by mapping them into the usual Kepler problem. It can also be employed to study some other inverse central-force problems and, in particular, it provides a proof of Bertrand's theorem.